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arXiv:1710.11316 · Physical review. B./Physical review. B

Spin- 12 Heisenberg antiferromagnet on the pyrochlore lattice: An exact diagonalization study

V. Ravi Chandra, Jyotisman Sahoo

Published Apr 12, 2018• 19 citations
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Abstract

We present exact diagonalization calculations for the spin-1/2 nearest-neighbor antiferromagnet on the pyrochlore lattice. We study a section of the lattice in the [111] direction and analyze the Hamiltonian of the breathing pyrochlore system with two coupling constants ${J}_{1}$ and ${J}_{2}$ for tetrahedra of different orientations and investigate the evolution of the system from the limit of disconnected tetrahedra (${J}_{2}=0$) to a correlated state at ${J}_{1}={J}_{2}$. We evaluate the low-energy spectrum, two and four spin correlations, and spin chirality correlations for a system size of up to 36 sites. The model shows a fast decay of spin correlations and we confirm the presence of several singlet excitations below the lowest magnetic excitation. We find chirality correlations near ${J}_{1}={J}_{2}$ to be small at the length scales available at this system size. Evaluation of dimer-dimer correlations and analysis of the nature of the entanglement of the tetrahedral unit shows that the triplet sector of the tetrahedron contributes significantly to the ground-state entanglement at ${J}_{1}={J}_{2}$.

DOI
10.1103/physrevb.97.144407

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We present exact diagonalization calculations for the spin-1/2 nearest-neighbor antiferromagnet on the pyrochlore lattice. We study a section of the lattice in the [111] direction and analyze the Hamiltonian of the breathing pyrochlore system with two coupling constants J1{J}_{1}J1​ and J2{J}_{2}J2​ for tetrahedra of different orientations and investigate the evolution of the system from the limit of disconnected tetrahedra (J2=0{J}_{2}=0J2​=0) to a correlated state at J1=J2{J}_{1}={J}_{2}J1​=J2​. We evaluate the low-energy spectrum, two and four spin correlations, and spin chirality correlations for a system size of up to 36 sites. The model shows a fast decay of spin correlations and we confirm the presence of several singlet excitations below the lowest magnetic excitation. We find chirality correlations near J1=J2{J}_{1}={J}_{2}J1​=J2​ to be small at the length scales available at this system size. Evaluation of dimer-dimer correlations and analysis of the nature of the entanglement of the tetrahedral unit shows that the triplet sector of the tetrahedron contributes significantly to the ground-state entanglement at J1=J2{J}_{1}={J}_{2}J1​=J2​.
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The phase diagram of a spin-1/2 J−J′−J2J-J'-J_2J−J′−J2​ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−J2J_1-J_2J1​−J2​ model on the square lattice with two different nearest-neighbor couplings J,J′J,J'J,J′ and may be also viewed as an array of coupled Heisenberg chains. The results suggest that the resonnating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B {\bf 67}, 024422 (2003)] for J2=0.5J′≪JJ_2=0.5J' \ll JJ2​=0.5J′≪J is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both side by a columnar dimer long range order. This columnar order could extends for J′→JJ'\to JJ′→J which correspond to the J1−J2J_1-J_2J1​−J2​ model.
Phase diagram of the antiferromagnetic J1J_1J1​-J2J_2J2​ spin-111 pyrochlore Heisenberg model
We study the phase diagram of the antiferromagnetic J1J_1J1​-J2J_2J2​ Heisenberg model on the pyrochlore lattice with S=1S=1S=1 spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second neighbor coupling we find that the non-magnetic ground-state phase of the nearest neighbor model extents up to $J_2/J_1 \sim 0.02 $ within DMRG, beyond which magnetic k=0\boldsymbol{k}=0k=0 long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime J2/J1∼0.035(8)J_2/J_1\sim 0.035(8)J2​/J1​∼0.035(8) which, together with the DMRG result, provides a strong argument for the existence of a small but finite non-magnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.
We study the spin-1/21/21/2 Heisenberg model on the triangular lattice with the nearest-neighbor J1>0{J}_{1}>0J1​>0, the next-nearest-neighobr J2>0{J}_{2}>0J2​>0 Heisenberg interactions, and the additional scalar chiral interaction ${J}_{\ensuremath{\chi}}({\stackrel{P\vec}{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\stackrel{P\vec}{S}}_{j})\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{S}}_{k}$ for the three spins in all the triangles using large-scale density matrix renormalization group calculation on cylinder geometry. With increasing J2{J}_{2}J2​ (${J}_{2}/{J}_{1}\ensuremath{\le}0.3$) and ${J}_{\ensuremath{\chi}}$ (${J}_{\ensuremath{\chi}}/{J}_{1}\ensuremath{\le}1.0$) interactions, we establish a quantum phase diagram with the magnetically ordered ${120}^{\ensuremath{\circ}}$, stripe, and noncoplanar tetrahedral phase. In between these magnetic order phases, we find a chiral spin liquid (CSL) phase, which is identified as a $\ensuremath{\nu}=1/2$ bosonic fractional quantum Hall state with possible spontaneous rotational symmetry breaking. By switching on the chiral interaction, we find that the previously identified spin liquid in the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model ($0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.15$) shows a phase transition to the CSL phase at very small ${J}_{\ensuremath{\chi}}$. We also compute the spin triplet gap in both spin liquid phases, and our finite-size results suggest a large gap in the odd topological sector but a small or vanishing gap in the even sector. We discuss the implications of our results on the nature of the spin liquid phases.